Method for controlling uniformity of thin films fabricated in processing systems

ABSTRACT

A method for spatial uniformity control in thin film processing is devised which is applicable to any film quality (thickness, composition, microstructure, electrical properties, etc.) as well as to all deposition systems (CVD, PVD, etch, ALD, etc.) where the substrate is rotated to improve uniformity of the deposited thin films. The technique is based on identifying the subspace of all deposition profiles on the stationary substrate that produce uniform films under rotation and then projecting a deposition profile to be controlled onto a sequence of uniformity—producing basis functions spanning that subspace to determine the Nearest Uniformity Producing Profile (NUPP). The process parameters as well as reactor design are optimized in order to minimize uniformity optimization criterion defined as a deviation of a produced deposition profile on the stalled substrate from the NUPP.

The subject utility Patent Application is based on the ProvisionalPatent Application Ser. No. 60/730,532 filed 26 Oct. 2005.

FIELD OF THE INVENTION

The present invention relates to thin film fabrication, and inparticular to fabrication of highly uniform thin films in processingsystems employing substrate rotation.

The overall concept of the present invention relates to a control ofspatial uniformity of thin films applicable to a broad range of filmquality parameters, e.g. thickness, composition, dopant levels,microstructure, electrical properties, morphology, etc., and forunlimited spectrum of thin film processing systems where a substrate(wafer) is rotated to improve thin film uniformity. One aspect of thenovel uniformity control technique is that it is based on a minimalnumber of physical assumptions, resulting in a technique applicable to awide range of thin film processing control, optimization, and designapplications. The uniformity control technique is applicable to, but notlimited to all systems using chemical vapor deposition (CVD), etch, PVD(physical vapor deposition), atomic layer deposition (ALD) and any otherthin film process employing a rotated substrate which gives thetechnique an extremely broad industrial impact.

Further, the present invention is related to a uniformity controltechnique which is based on identifying a subspace of all depositionprofiles on the stationary (stalled) substrate that produces uniformfilms under rotation, by projecting a deposition profile to becontrolled onto a sequence of uniformity-producing basis functions, anddetermining the “Nearest Uniformity Producing Profile” (NUPP). Thismathematical criterion depends only on the geometrical characteristicsof the thin film processing system. A deviation of the real profile ofthe thin films formed from the NUPP is minimized substantially to zeroin the present uniformity optimization technique in order to attain thespatial uniformity of the thin films through adjustment and control ofthe process and/or design parameters of thin film processing systems.

BACKGROUND OF THE INVENTION

Rapid evolution of material systems and continued tightening of qualitycontrol constraints for thin-film manufacturing processes insemiconductor and other (e.g., optical coating) industries pose a numberof challenges to equipment design, giving rise to a wide range ofreactor systems designed to reduce spatial non-uniformity of depositionthickness, composition, microstructure and other quality parameters ofthin films. In some manufacturing processes, the use of substrate(wafer) rotation is integral to achieving acceptable film propertiesacross the substrate. In Chemical Vapor Deposition (CVD) systemscommonly used for semiconductor processing numerous reactor designs makeuse of wafer rotation, such as

1. cross-flow reactor designs shown in FIG. 1A, where gas flows througha tube or duct-shaped reactor chamber over a wafer and exhausts oppositethe gas inlet where wafer rotation is used to reduce cross-flowdeposition non-uniformities and depletion effects in the direction offlow;

2. cylindrical reactors shown in FIG. 1B, in which gas flows from ashower head over a wafer and exhausts from the bottom, where waferrotation is used to eliminate any residual angular non-uniformities inthe reactor design; and

3. planetary reactors shown in FIGS. 1C-1D, where gas flows radiallyoutward from a central feed point over a susceptor containing multiplewafers. As the gas flows over the hot susceptor, thermal decompositionand gas phase and surface reactions take place, some of which result infilm deposition of the desired material, while other reactions lead tothe formation of gas-phase adducts and deposits on reactor walls andother surfaces.

Due to depletion and the manner in which decomposition reactions takeplace in this reactor geometry, radial flow designs inherently producenon-uniform deposition patterns with respect to the radial coordinate ofthe reactor. In an effort to overcome this limitation, some reactordesigns incorporate a substrate planetary motion mechanism as part ofthe susceptor assembly to compensate for the depletion and other radialand azimuthal variations in CVD reactor systems as shown in FIG. 1C. Inthese reactors, the relatively large susceptor 10 rotates around itscenter point and each wafer 12 rotates independently of the susceptor.This combination of rotating motions results in points on each wafertracing out a cycloid-like pattern partially compensating for thenon-uniform deposition profile. Typically, these reactors can beoperated with rotating wafers or stalled (non-rotating) wafers. Thisdesign has the effect of eliminating reactor-induced angularnon-uniformity generators through susceptor rotation. Wafer rotation isused to reduce the intrinsic (and completely unavoidable) effect of gasphase reactant decomposition and precursor depletion in the gas phase.

The physical and electrical properties of SiC and group-III nitrides(e.g., AlN, GaN, InN, and their alloys) make these materials ideal forhigh-frequency, high-power electronic devices as well as optoelectronicapplications. For example, gallium nitride (GaN), a compoundsemiconductor material, has shown potential in electronic andoptoelectronic devices over the past few years due to its wide-bandgapand high breakdown field properties. GaN has a direct bandgap of 3.4 eVmaking it suitable for manufacturing light emitting diodes (LEDs)capable of emitting light of any wavelength between blue and ultraviolet(UV) when alloyed with indium (In) and aluminum (Al). In addition,GaN-based devices are used for high-frequency and/or high-powerapplications including aircraft radar electronics.

Metalorganic vapor phase epitaxy (MOVPE) is the principal method used togrow single-crystalline layers of this material. Currently,manufacturers of gallium nitride devices use both commercial andcustom-built reactor designs. The wide range of reactor designsindicates a lack of a coherent framework on how to design galliumnitride reactors for optimal single wafer and multiple-wafer production.As a result, significant research from both academic and industriallevels has enhanced manufacturing technology considerably within thepast decade.

Despite ongoing research in this area, an unambiguous understanding ofthe physical and chemical mechanisms governing the deposition process isstill lacking. The difficulties in achieving this understanding to acertain extent can be linked to the complex intrinsic chemistry of thedeposition process, the knowledge of which currently is incomplete. Alarge number of gas phase and surface phase reactions resulting from theextreme conditions necessary for gallium nitride growth have beenextensively studied by many researchers. As a result, a number ofchemical mechanisms describing important gas phase and surface phasereactions during GaN growth have been reported in the literature. Thoughmost of these mechanisms present similar reaction pathways, thedistinguishing factors are the individual rate parameters. In addition,some research groups assume significant gas phase reactions, whereasothers assume gas phase reactions play no role in film depositionkinetics. A consensus of a definitive kinetic model describing galliumnitride growth has yet to be reached.

Gas phase gallium nitride chemistry may be visualized as consisting oftwo competing routes: an a) upper route and b) lower route. The upperroute is more commonly referred to as the adduct formation pathway,whereas, the lower route refers to the thermal decomposition pathway ofTMG (Trimethylgallium). Each pathway is responsible for producing anarray of chemical species that may eventually participate in GaNdeposition. The primary gas phase reaction is the spontaneousinteraction between commonly used precursors, trimethylgallium((CH₃)₃Ga) and ammonia (NH₃), to form stable acid-Lewis base adducts.Adduct formation is a ubiquitous problem during MOVPE of GaN and hasbeen widely studied. Upon formation, these adducts may condense on coldsurfaces inside the reactor system. For this reason, the formation ofthese adducts is believed to degrade film quality, uniformity, andconsume the feed stream of organometallic sources.

Consequently, numerous research groups have designed reactor systems, inparticular gas delivery systems, with the intent to minimize precursorinteractions. The most common approach has been to use separateinjectors to reduce any premature mixing of the precursors. Reactorsystems of this type have been developed by SUNY/Sandia/Thomas Swannresearchers to illustrate a connection between gas phase reactions andfilm-thickness uniformity. It should be noted that while these designsare able to suppress reactions in the gas delivery system, completemixing of the precursors must take place close to the wafer surface toachieve uniform film thickness.

More novel approaches to the optimization of GaN CVD are based onoptimization of two objective functions that span multiple length scaleswhich are performed simultaneously to maximize thickness uniformity(macroscopic objective) and minimize surface roughness across wafersurface (microscopic objective). Additionally, a strategy for nonlinearprogramming problems that involved PDE models has been developed andapplied to a detailed GaN CVD model where the objective was to optimizeoperating conditions that produced thin films of GaN with spatiallyuniform thickness.

While a number of simulation-based optimization studies have beenperformed on existing reactor systems, none of them however haveaddressed a fundamental question of whether the non-uniform depositionprofiles exist in the reactor radial coordinate which produce perfectlyflat deposition profiles on the rotated wafers.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a method forcontrolling spatial uniformity of thin films which is universallyapplicable to any quality parameter of thin films (thickness,composition, microstructure, morphology, electrical properties, etc.)fabricated in a film processing system (including CVD, etch, PVD, ALD,etc.) with substrate rotation.

It is a further object of the present invention to identify non-uniformdeposition profiles of thin films formed on stalled substrates whichproduce uniform thin film profiles upon the rotation of the substratesand to adjust fabrication process and reactor design parametersaccordingly to obtain such identified non-uniform deposition profiles onthe stalled substrate in the optimized fabrication process and reactor.

It is an additional object of the present invention to develop anunambiguous uniformity criterion, e.g. a “Nearest Uniformity ProducingProfile” (NUPP) independent of a nature of thin films or processingsystems, which permits to formulate an optimization criterion defined asthe minimization of deviation of a produced deposition profile from theNUPP.

In the novel method for controlling the uniformity of produced thinfilms, a numerical analysis procedure begins by defining a sequence oforthogonal, complete functions over the region of a reactor where wafersare located. Subsequently, each orthogonal complete function isprojected, in sequence, onto the wafers physical domain, and therotation-averaged transformed functions are determined. Using theSingular-Value Decomposition (SVD) method to define an orthogonal basisfor the rotation-averaged functions, it is further determined whichcombinations of the new basis functions best approximate a flat filmprofile. This approach is used to find a sequence of modes, each ofwhich generates perfectly flat profiles under rotation.

Due to the fact that linear combinations of the final sets of modesβ_(n) generate flat profiles, these β_(n) are used to generate a usefulbasis onto which a deposition profile can be projected to immediatelydetermine whether the particular profile will generate uniform filmsunder rotation, and if it does not, predict the shape of the “nearest”profile that does. Likewise, these modes β_(n) can be used as a part ofan efficient mechanism of optimizing the deposition process foruniformity.

Specifically, the method of the present invention for improvinguniformity of thin films formed in a thin film processing systemcomprises the steps of:

providing a thin film processing system including at least one substrateoperated in a stalled substrate mode and in a rotating substrate mode,

providing a computation system operationally coupled to said filmprocessing system,

identifying a first distribution profile of a parameter underinvestigation of a thin film formed on the stalled substrate whichproduces a uniform second distribution profile of the thin film uponsubstrate rotation in the rotating substrate mode;

computing a subspace of basis functions β_(n) corresponding to the firstdistribution profile;

formulating a uniformity optimization criterion defined as a deviationof a deposition profile of the parameter under investigation from thesubspace of the basis functions β_(n); and

optimizing the thin film processing system to minimize the uniformityoptimization criterion substantially to zero.

To formulate the uniformity optimization criterion, a Nearest UniformityProducing Profile (NUPP) N_(u)(r,θ) is formed which is based on thesubspace of basis function β_(n), wherein

${N_{u}\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{N}\;{{\beta_{n}\left( {r,\theta} \right)}{\int_{0}^{2\pi}{\int_{0}^{r_{\omega}}{{\delta\left( {r,\theta} \right)}{\beta_{n}\left( {r,\theta} \right)}r\ {\mathbb{d}r}\ {\mathbb{d}\theta}}}}}}$

wherein β_(n)(r,θ) is the basis function corresponding to the firstdistribution profile,

δ(r,θ) is the deposition profile of the parameter under investigation,and

r and θ are parameters of the substrate physical domain ω(r,θ).

The uniformity optimization criterion S(r,θ)=δ(r,θ)−N_(u)(r,θ) iscalculated at a predetermined region of the thin film.

A film profile under investigation Δ(x, y) is projected onto thephysical domain ω(r,θ) of the substrate, wherein Δ(x, y) represents theparameter under investigation of said thin film, thus defining thedeposition profile δ(r,θ) of said parameter, wherein

${\delta\left( {r,\theta} \right)} = {\sum\limits_{i,j}^{I,J}{a_{i,j}{p_{i,j}\left( {r,\theta} \right)}}}$

wherein p_(i,j)(r,θ) are representations of said deposition profileδ(r,θ) over said physical domain ω(r,θ) of the substrate, and

α_(i,j) are contribution coefficients.

Further, rotation—averaged functions δ(r,θ) are determined whichcorrespond to said δ(r,θ),

wherein

${{\overset{\_}{\delta}\left( {r,\theta} \right)} = {{R\;{\delta\left( {r,\theta} \right)}} = {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{\alpha_{i,j}(r)}}}}},$

wherein R is a rotationally—averaging operator, and

α_(i,j)(r) are representations of rotationally averaged functionscorresponding to p_(i,j)(r,θ).

A subset {{circumflex over (p)}}_(n=1) ^({circumflex over (n)}) of allsaid p_(i,j)(r,θ) is selected which corresponds to non-uniform andnon-zero {circumflex over (α)}_(i,j)(r) from α_(i,j)(r) for furthercomputation to define uniformity producing modes {β_(n)}_(n-0) ^(N) ofthe subspace span {{circumflex over (p)}} corresponding to all non-zeroprofiles of the parameter under investigation of the thin film formed onthe substrate in the stalled substrate mode producing uniform thin filmupon rotation of the substrate.

The present invention is also directed to a thin film processing systemfor fabrication of thin films of high uniformity. The system includes:

a processing chamber, and

a computation system operationally coupled to the processing chamberwhich includes a thin film uniformity optimization and control unit.

The uniformity optimization and control unit comprises:

(a) a first unit identifying at least one first distribution profile ofat least one parameter of a thin film formed on a stalled substratewhich produces a uniform second distribution profile of the parameterupon substrate rotation;

(b) a second unit computing a subspace of basis functions β_(n)corresponding to the first distribution profile generating a uniformsecond distribution profile;

(c) a third unit forming a Nearest Uniformity Producing Profile (NUPP)N_(u)(r,θ) based on the subspace of basis function β_(n), where

${N_{u}\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{N}\;{{\beta_{n}\left( {r,\theta} \right)}{\int_{0}^{2\pi}{\int_{0}^{r_{\omega}}{{\delta\left( {r,\theta} \right)}{\beta_{n}\left( {r,\theta} \right)}r\ {\mathbb{d}r}\ {\mathbb{d}\theta}}}}}}$

wherein β_(n)(r,θ) are said basis functions corresponding to the firstdistribution profile,

δ(r,θ) is a deposition profile of said parameter under investigation,and r and θ are parameters of the substrate physical domain ω(r,θ),

(d) a fourth unit formulating a uniformity optimization criterionS(r,θ)=δ(r,θ)−N_(u)(r,θ) defined as a deviation of the depositionprofile δ(r,θ) of the parameter from the NUPP N_(u)(r,θ) at apredetermined region of the thin film; and

(e) a fifth unit creating the uniformity optimization strategy foradjusting the thin film processing system to minimize the uniformityoptimization criterion.

These and other features and advantages of the present invention will beclear from further description of the preferred embodiment inconjunction with the Drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1C are schematic representations of cross-sectional views ofthree CVD conventional reactor designs, all featuring wafer rotation:cross flow (duct) single wafer reactor (FIG. 1A), single wafercylindrical reactor (FIG. 1B), multiple wafer radial flow planetaryreactor system (FIG. 1C);

FIG. 1D is a representation of the physical domain of the susceptor andwafers in the multiple wafer radial flow planetary reactor system ofFIG. 1C;

FIG. 2 are graphs representing the P_(m), m=1, 2, 3, 4 and thecorresponding α_(m) for the reactor geometry shown in FIGS. 1C-1D;

FIG. 3 is a graphical representation of singular values of σ_(m)generated when computing the span of the rotationally averaged shiftedLegendre polynomials;

FIG. 4 is a graph representing a plurality of β_(n) functions;

FIG. 5 is a diagram representative of the first four β_(n) modesprojected onto a stalled wafer;

FIGS. 6A-6C are graphs representing gas velocity, temperature andcomposition profiles, respectively. The wafers are located in the regionof FIG. 6C between 0.06 m≦R≦0.14 m;

FIG. 7 is a diagram representing wafer deposition profile δ, equivalentto Δ between 0.06 m≦R≦0.14 m, the resulting profile upon rotation dataδ, the nearest uniformity generating profile (NUPP) ƒ and NUPP profileupon rotation ƒ. C_(d) is computed as a difference between δ and ƒ atR=0 (R=0.1 m);

FIG. 8A is a diagram showing the C_(d) and derivatives of δ versus inletgas velocity;

FIGS. 8B and 8C are the diagrams of deposition profile δ, equivalent toΔ between 0.06 m≦R≦0.14 m, the resulting profile upon rotation δ, NUPP ƒand NUPP profile upon rotation ƒ at ν₀=30 m/s and ν₀=40 m/s,respectively;

FIGS. 9A-9C are representations of reactor/wafer quadrature gridgeometry over the deposition and wafer domains for cross flow,cylindrical and planetary reactors, respectively;

FIG. 10 represents a plurality of projections p_(i,j) of a complete setof basis functions over the deposition domain of the wafer;

FIGS. 11A and 11B are representations of an example of p^(o) (FIG. 11A)and {circumflex over (p)} (FIG. 11B);

FIG. 12 is a representation of the β_(n) modes;

FIGS. 13A-13D illustrate the application of the proposed uniformityoptimization technique to a planetary CVD reactor simulation. In FIG.13A, a reactor concentration profile curves X_(A), X_(B) and X_(C)correspond to an inlet gas velocity of 20 m/s. The resulting depositionprofile over the wafer is shown in FIG. 13B, and the NUPP is shown inFIG. 13C. In FIG. 13D, the current deposition profile and NUPP profilesare superimposed with an arrow indicating the direction of gas flow;

FIGS. 14A-14D illustrate a diagram representative of reactorconcentration profile curves X_(A), X_(B), and X_(C) corresponding to anoptimized inlet gas velocity of 35.04 m/s. The resulting depositionprofile over the wafer is shown in FIG. 14B, the NUPP is shown in FIG.14C, and the current deposition profile and NUPP profiles aresuperimposed in FIG. 14D with an arrow indicating the direction of gasflow;

FIG. 15 is a simplified schematic of the processing system of thepresent invention for fabrication of highly uniform thin films;

FIGS. 16A-16B represents schematically two modes of operation of thereactor;

FIGS. 17 and 18 are flow chart diagrams of the sequence of stepsunderlying the operation of the thin film processing system of thepresent invention; and

FIG. 19 is a schematic representation of a reactor susceptor designfeaturing a recess that contains “a monitoring strip” for determiningthe stalled wafer profile without sacrificing the wafer.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A fundamental question of finding the deposition profiles which produceperfectly flat deposition profiles on the rotated wafers (substrates) isexamined herein, and an approach is presented for identifying thesubspace of all stalled wafer profiles that produce uniform films uponrotation. The basis for this subspace is used to identify the NearestUniformity Producing Profile (NUPP), a criterion shown to be useful foroptimizing thin film uniformity in planetary reactor systems. Theconcept of this uniformity producing subspace is broadened herein toinclude a substantially unlimited spectrum of reactor geometries for theimplementation of NUPP-based uniformity optimization techniques.

The unambiguous uniformity criterion “Nearest Uniformity ProducingProfile” (NUPP) is developed which is the basis for formulating anoptimization criterion defined as the minimization of the distancebetween the NUPP and the profile of the deposited thin film. Thedistance may be minimized as part of a process recipe developmentprocedure in a run-to-run control system or as part of reactor geometryadjustment. Thus formulated optimization criterion depends only on thereactor and wafer dimensions, and thus may be universally applied to anydistributed film quality to be controlled and to any reaction andmaterial system. For example, this approach to uniformity control can beapplied to film thickness and composition (e.g., dopant level) control,as well as to the control of other film characteristics. Theoptimization technique of the present invention is also applicable toother applications, such as, for example, ion-beam etching tools whereplanetary motion is used to compensate for beam divergence, as well asin optical coating processes where planetary deposition systems are usedto reduce the effect of non-uniformities in the coating plume.

Uniformity Modes

The development of the uniformity analysis approach is initiated bydefining the physical domains of the reactor and wafer. Using thenotation of FIG. 1D, the susceptor and wafer domains Ω and ω,respectively, their respective coordinate systems consist ofΩ:R_(s)−r_(w)≦R≦R_(s)+r_(w) and ω:0≦r≦r_(w), 0≦θ≦2π, or in Cartesiancoordinates, ω:(x, y) such that x²+y²≦r_(w) ². Δ(R) is defined as theone dimensional deposition profile of a quality parameter of interest ofthe deposited thin film over the susceptor radial coordinate. Thedeposition profile Δ(R) is assumed independent of the azimuthalcoordinate due to susceptor rotation. δ(x, y) or δ(r,θ) is the twodimensional profile that results when Δ(R) is projected onto the stalled(non-rotating) wafer physical domain. Therefore, for x=0, the depositionprofiles are related byδ(0,y)=Δ(y+R _(s))  (Eq. 1)The relationship between the coordinate systems isx ²+(y+R _(s))² =R ²  (Eq. 2)so thatδ(x,y)=Δ(√{square root over (x ²+(y+R _(s))²)})  (Eq. 3)δ(r,θ)=Δ(√{square root over (r ² cos² θ+(r sin θ+R _(s))²)})  (Eq. 4)

Having defined the film profile δ(r, θ) as corresponding to the stalledwafer deposition pattern, wafer rotation will eliminate the θ dependenceby averaging the film quality over θ giving

$\begin{matrix}{{\overset{\_}{\delta}(r)} = {\frac{1}{2\pi}{\int_{0}^{2\pi}{\Delta\left( {\sqrt{\left. {{r^{2}\cos^{2}\theta} + \left( {{r\;\sin\;\theta} + R_{s}} \right)^{2}} \right)}\ {\mathbb{d}\theta}} \right.}}}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$

It is physically reasonable to assume the deposition profiles Δ(R) aresmooth, continuous, bounded functions. Therefore, any profile Δ(R) overthe wafer segment domain R∈Ω can be represented by an expansion in termsof a complete basis function sequence in L²(Ω). For example, if P_(m)(R)are the shifted Legendre polynomials defined on Ω, then

$\begin{matrix}{{\Delta(R)} = {\sum\limits_{m = 0}^{\infty}{a_{m}{P_{m}(R)}}}} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$with P₀=1. Therefore,

$\begin{matrix}{{\overset{\_}{\delta}(r)} = {\sum\limits_{m = 0}^{\infty}{a_{m}\frac{1}{2\pi}{\int_{0}^{2\pi}{P_{m}\left( {{\sqrt{\left. {{r^{2}\cos^{2}\theta} + \left( {{r\;\sin\;\theta} + R_{s}} \right)^{2}} \right)}\ {\mathbb{d}\theta}} = {\sum\limits_{m = 0}^{\infty}{a_{m}{\alpha_{m}(r)}}}} \right.}}}}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$in ω.

Since P₀ is constant with respect to R, α₀ will likewise be constant.Therefore, this first mode corresponds to the most obvious depositionprofile that would produce uniform films on the rotated wafers: Ifα_(m)=0 for m=1, 2, . . . and α₀≠0, then δ(r) is flat because Δ islikewise flat. Therefore, P₀, and the corresponding α₀, are removed fromthe analysis since the goal is to find non-uniform Δ(R) profiles thatproduce uniform δ(r) profiles under rotation. The P_(m), m=1, 2, 3, 4and the corresponding α_(m) are shown in FIG. 2 for the reactor geometryshown in FIG. 1.

Having eliminated the purely uniform mode from the problem formulation,the goal stated above may be written in terms of the minimizationproblem

$\begin{matrix}{\min\limits_{a_{m}}{{{\sum\limits_{m = 1}^{M}{a_{m}{\alpha_{m}(r)}}} - c}}} & \left( {{Eq}.\mspace{14mu} 8} \right)\end{matrix}$where c is a constant with respect to r and c≧0; M is a truncationnumber set sufficiently large to capture the smallest anticipatedfeature size to be controlled in the film. For the cases c>0, the modeamplitude coefficients am can be scaled by c, and it is legitimate toset c=1 in (Eq. 8) without loss of generality.

Defining the array Q of inner products over ω,Q_(ij)=

α_(j)α_(j)

i=1, . . . , M  (Eq. 9)then Q=UΣV^(T)  (Eq. 10)where the Singular Value Decomposition (SVD) produces U^(M×M) containingthe orthogonal singular vectors and Σ the singular values. From theseproducts of the SVD, an orthogonal set of basis functions {α_(n)^(υ)}_(n=1) ^(N) is defined using

$\begin{matrix}{\alpha_{n}^{\upsilon} = {\sum\limits_{m = 1}^{M}{U_{m,n}{\alpha_{m}(r)}}}} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$for the space spanned by the {α_(n)}_(m=1) ^(M).

Typically, the singular values of Q used to define the α_(n) ^(υ) appearas those shown in FIG. 3. The set of relatively small magnitude σ_(m)indicate that there are fewer than M α_(n) ^(υ) necessary to span thespace of the sequence {α_(m)}_(m=1) ^(M). The breakpoint separatingsignificant from relatively small singular values depends on the valueof M and is difficult to define rigorously; therefore, Eq. 8 is solvedby computing

$\begin{matrix}{\in_{N}{= {\min\limits_{a_{j}^{\upsilon}}{{{\sum\limits_{j = 1}^{N}{a_{j}^{\upsilon}\alpha_{j}^{\upsilon}}} - 1}}}}} & \left( {{Eq}.\mspace{14mu} 12} \right)\end{matrix}$for all N≦M and by choosing N such that ∈_(N) is minimized (since theα_(j) ^(υ) are orthonormal, the α_(j) ^(υ) are found by projecting theconstant function 1 onto the α_(j) ^(υ)). That the optimal value of Ngenerally is less than M because for small values of N the expansion inα_(j) ^(υ) produces a poor approximation of 1 because of an insufficientnumber of basis functions, while a value of N approaching M will resultin the expansion including the α_(j) ^(υ) that correspond to avanishingly small σ_(j), reducing the accuracy of the projectionoperation. This calculation leads to the definition of the firstnon-flat deposition mode that generates uniform films when rotated:

$\begin{matrix}{{\beta_{1}(R)} = {\sum\limits_{m = 1}^{M}{b_{m}{P_{m}(R)}}}} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$where β₀=constant and

$\begin{matrix}{b_{m} = {\sum\limits_{n = 1}^{N}{U_{m,n}a_{n}^{\upsilon}}}} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

The procedure described above produces a single β_(n) profile.Sequentially removing the lower-frequency α_(j) ^(υ) from the problem offinding the α_(j) ^(υ) that minimizes ∈_(N) forces the minimizationprocedure to find a progression of modes correspondingly increasing infrequency. Therefore, additional β_(n) modes may be produced byminimizing the sequence of problems

$\begin{matrix}{\in_{N}{= {\min\limits_{a_{j}^{\upsilon}}{{{{\sum\limits_{j = n}^{N}{a_{j}^{\upsilon}\alpha_{j}^{\upsilon}}} - 1}}.}}}} & \left( {{Eq}.\mspace{14mu} 15} \right)\end{matrix}$for n=2, 3, . . . to obtain the corresponding β_(n). These modes,corresponding to the reactor geometry shown in FIGS. 1C-1D afterorthonormalization, are seen in FIG. 4. The computations are carried outusing the quadrature grid based weighted residual method techniquespresented in Adomaitis R. A., “Objects for MWR” Computers & ChemicalEngineering, 2002, 26 (7-8), pp. 981-998.

Projecting the β_(n) onto the stalled wafer domain ω, the first fournon-flat deposition profiles that produce uniform profiles when rotatedare observed, as shown in FIG. 5. Because these modes define at leastpart of the basis for the uniformity producing subspace of all possibledeposition profiles, any linear combination of the β_(n) modes producesa uniform deposition profile under rotation. These modes β_(n) dependonly on the wafer and susceptor geometry, and therefore are universal innature. The numerical techniques presented in this applicationaccurately capture between several and tens of these modes beforenumerical difficulties in accurately minimizing (Eq. 15) areencountered.

Deposition Profile Model

To illustrate the utility of the β_(n) modes in a CVD processoptimization application, a representative gas-phase thermaldecomposition of a deposition precursor species A through a sequence ofirreversible reactions (B→C) is considered:A→B→Cwith the deposition rate k_(d) being a function of gas phase molefraction of Species C:Δ(R)=k _(d) x _(C)  (Eq. 16)The deposition rate k_(d) is a constant set to k_(d)=1 film thicknessunits/time. Typically, the precursor is fed to the reactor system nearroom temperature and at a low concentration level in a carrier gas thatplays little or no role in the gas phase or deposition reaction.Therefore, a material balance for each of the three reactive gas phasespecies under the assumption of the ideal gas law can be written as

$\begin{matrix}{{\frac{1}{R}\frac{\mathbb{d}}{\mathbb{d}R}\left( {\frac{1}{T}R\;\upsilon\; x_{A}} \right)} = {{- \frac{1}{T}}{k_{1}(T)}x_{A}}} & \left( {{Eq}.\mspace{14mu} 17} \right) \\{{\frac{1}{R}\frac{\mathbb{d}}{\mathbb{d}R}\left( {\frac{1}{T}R\;\upsilon\; x_{B}} \right)} = {{{- \frac{1}{T}}{k_{2}(T)}x_{B}} + {\frac{1}{T}{k_{1}(T)}x_{A}}}} & \left( {{Eq}.\mspace{14mu} 18} \right) \\{{\frac{1}{R}\frac{\mathbb{d}}{\mathbb{d}R}\left( {\frac{1}{T}R\;\upsilon\; x_{C}} \right)} = {{{- \frac{1}{T}}{k_{3}(T)}x_{C}} + {k_{d}x_{C}}}} & \left( {{Eq}.\mspace{14mu} 19} \right)\end{matrix}$over Ω, subject to initial conditions x_(i)(R=R₀)=x_(i0). The continuityequation for the total gas molar flow rate gives

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}R}\left( {v\; p\; R} \right)} = {{0\mspace{14mu}{or}\mspace{14mu} v} = \frac{v_{0}R_{0}T}{T_{0}R}}} & \left( {{Eq}.\mspace{14mu} 20} \right)\end{matrix}$and this result is used to simplify the species balance equations.

Finally, assuming the heat capacity and the gas/susceptor heat transfercoefficient are independent of temperature, the gas phase thermal energybalance gives

$\begin{matrix}{{\frac{1}{R}\frac{\mathbb{d}}{\mathbb{d}R}\left( {v\; R} \right)} = {h\left( {T_{s} - T} \right)}} & \left( {{Eq}.\mspace{14mu} 21} \right)\end{matrix}$for the case where the susceptor is at a constant temperature T_(s) andthe reactor roof is perfectly insulating.

For this system, the concentration profiles are computed forR₀≦R≦R_(sus) with R₀=0.01 m corresponding to the mixing regionsurrounding the central feed tube, susceptor diameter R_(p)=0.2 m,satellite wafer centerline radius R_(s)=0.1 m, and wafer radiusr_(ω)=0.04 m. Spacing between the susceptor (wafer top) surface and thereactor roof is set at 0.02 m, and the initial value of ν₀=20 m/s.

Typical results are shown in FIGS. 6A-6C where the axial mean of the gasradial velocity, gas temperature, and the concentration profiles of eachchemical species are plotted as a function of R. In these simulationresults, the rapid decrease in gas radial velocity is observed due tothe radial flow geometry, the rapid gas heating inside the reactor, andthe decomposition of species A to B followed by the quick decompositionof the deposition species C. The peak concentration of species C isobserved near the wafer centerline.

In the analysis presented in the following paragraphs it is shown thatthis species concentration profile results in a highly non-uniformradial profile when the wafer is rotated. Using an optimizationtechnique based on the β_(n), the change in operating conditions isdetermined which is necessary to improve the deposition uniformity.

Optimization Criterion Formulation

For the planetary reactor deposition system simulation underconsideration, the “nearest” deposition rate profile ƒ that generates auniform film upon rotation ƒ is shown in FIG. 7. This optimal profile ƒ,henceforth referred to as the “Nearest Uniformity Producing Profile”(NUPP), is computed as the projection of the deposition profile Δ(R)onto the β_(n):

$\begin{matrix}{f = {\sum\limits_{n - 1}^{N}{\left( {{\Delta(R)},\beta_{n}} \right)\beta_{n}}}} & \left( {{Eq}.\mspace{14mu} 22} \right)\end{matrix}$where the inner product is defined on Ω.

In this example, the NUPP and Δ are, for the most part, increasingfunctions with R for R∈Ω, and both curves have a peak value within Ω.The distance C_(d) between the wafer centerline values of Δ and ƒ ispositive and has a value that is a significant fraction of the mean filmthickness itself.C _(d) =Δ−ƒ  (Eq. 23)Based on the subject computation, physical intuition suggests increasingthe inlet gas velocity to “push” the deposition peak towards the reactoroutlet (to the right of the diagram presented in FIG. 7).

Motivated by this physical insight into a potential route to improvingthickness uniformity, solutions are computed for a range of ν₀ and theresults are presented in FIG. 8 a, illustrating a plot of C_(d) vs. ν₀.Additionally, the solutions and their NUPPs are presented in FIGS. 8 band 8 c for two additional representative values of ν₀=30 m/s and ν₀=40m/s, respectively. An important observation thus may be made from thissequence of simulations: when ƒ and Δ intersect at the wafer centerR=R_(s) (R_(s)=0.1 in this diagram), C_(d)=0 and the uniformity in thecenter region of the wafer is produced. Of course, wafer rotation andthe assumption that deposition profiles are in L²(Ω) guarantees d δ/dr=0at r=0 regardless of the operating conditions. However, all derivativesof the rotated NUPP are zero and so the effect of the intersection ofthe deposition curve and the NUPP is to drive the second derivative of δwith respect to r to zero, resulting in a very flat wafer profile in theneighborhood of the wafer center. It is clear from FIG. 8 a that d²δ/dr²>0 when C_(d)<0 and d² δ/dr²<0 when C_(d)>0 for this system.

These observations lead to an unambiguous design criterion of C_(d)=0for improved wafer uniformity in the central region of the wafer, whichis in many cases, the region where uniformity is most desired. In thecase of simulation based process optimization, operating conditionssatisfying this condition are easily determined simultaneously with thesimulator solution using a Newton-Raphson technique. For this example,it was found that the optimal inlet velocity ν₀=35.04 m/s, e.g., thepoint marked in FIG. 8 a.

It is important to note that the C_(d)=0 condition is not simplyequivalent to determining the operating conditions where d² δ/dr²=0 atthe wafer center. In practice it is difficult to obtain accurate valuesof derivatives of wafer measurements because of noise and other sourcesof error. The wafer metrology data is projected onto a relatively smallnumber (e.g., four in this case) of the least oscillatory β_(n) modesand the projection operation has a natural noise-filtering effect whichimproves the accuracy with which the criterion is evaluated.

Furthermore, in addition to the potential physical insight into processoptimization revealed in this analysis approach, the relative slope ofthe intersecting Δ and ƒ curves at r=0 when C_(d)=0 may determine howquickly the films become non-uniform in the direction away from thewafer center. Likewise, the overall slope of the intersecting Δ and ƒcurves may determine how robust the film uniformity is to model errorsand process unknowns.

In the above paragraphs, a new approach to film uniformity control ispresented for planetary CVD systems. The approach to determining the“nearest” optimal profile has potential in interpreting measurementsfrom stalled wafer experiments to provide guidance on process conditionmodifications that would lead to improved uniformity. Experimental workhas been conducted to implement these concepts to improve theperformance of a commercial CVD reactor system using a run-to-runcontrol criterion based on driving C_(d) to zero. To implement thisuniformity optimization approach to a run-to-run control framework, thereactor is run with a single stalled wafer that is sacrificed to makethe measurements necessary to tune the tool to the desired operatingpoint and to maintain its performance against drift and otherdisturbances.

Determining the NUPP function ƒ is shown to lead to real physicalinsight on how reactor operating conditions should be adjusted toimprove uniformity. Additionally, NUPP functions with segments that arephysically infeasible (such as negative deposition rate profiles) mayindicate that the reactor system is operating “far” from a feasibleuniformity profile.

The uniformity control criterion C_(d) is a step towards a new, abstractand universal equipment design approach, where CVD systems may bedesigned to maximize the likelihood of producing films in thisuniformity subspace: since (n−1)-dimensional surfaces may be found whereC_(d)=0 in a n-dimensional design parameter space, it is possible toseparate the design variables into a distinct subset to be used tooptimize other processing criteria and to use the remainder to satisfythe C_(d) minimization criterion.

Furthermore, it is possible, using the approach of the present inventionto generate reactor designs that minimize the sensitivity of satisfyingthis criterion to design model by choosing design points on the C_(d)=0surface where satisfaction of this criterion is least sensitive to themost uncertain elements of the process model used for reactor design.The broadening of the applicability of the NUPP criterion from theplanetary reactor system, shown in FIGS. 1C-1D, to a wider range ofsystems, and examination in more detail of the structure of theuniformity-producing subspace, its implications in developing uniformitycontrol techniques, and improved process characteristics and reactordesigns are presented in the following paragraphs.

For any reactor configuration, reactor Ω(x,y) and wafer ω(r,θ) physicaldomains are defined, as shown in FIGS. 9A-9C showing cross-flow,cylindrical and planetary reactor geometries, respectively. Film growthtakes place in reactor physical domain Ω. A property Δ(x, y) underinvestigation which is desired to be spatially uniform is defined in atleast a portion of this domain Ω. Given the complete basis functionsequences {φ_(i)(x)}i^(∞)=1 and {ψ_(j)(y)}j^(∞)=1 (such as sequencing ofpolynomial functions, or Fourier function sequencing), this filmproperty is represented by

$\begin{matrix}\begin{matrix}{{\Delta\left( {x,y} \right)} = {{\sum\limits_{i,{j = 1}}^{\infty}{a_{i,j}{\phi_{i}(x)}{\psi_{j}(y)}}} =}} \\{= {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{\phi_{i}(x)}{\psi_{j}(y)}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 24} \right)\end{matrix}$where the finite truncation numbers I, J can be used because ofdiffusion and other physical phenomena that limit the maximum lengthscale that must be resolved by the basis function expansion, and whereinα_(i,j) are coefficients representing contribution of each basisfunction φ_(i)(x) and ψ_(j)(y) into Δ(x, y). Defining P as the operatorthat projects Δ onto ω,δ(r,θ)=PΔ(x,y)  (Eq. 25)the stalled wafer deposition profiles δ(r,θ) are found

$\begin{matrix}\begin{matrix}{{\delta\left( {r,\theta} \right)} = {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}P\;{\phi_{i}(x)}{\psi_{j}(y)}}}} \\{= {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{p_{i,j}\left( {r,\theta} \right)}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 26} \right)\end{matrix}$Representative p_(i,j) which are the basis functions over the wafer ωwhich are used to represent the true profile δ(r,θ) are shown in FIG.10.

Further, defining R as the operator that determines therotationally-averaged δ profiles of the stalled wafer profiles δ:

$\begin{matrix}\begin{matrix}{{\overset{\_}{\delta}(r)} = {R\;{\delta\left( {r,\theta} \right)}}} \\{= {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}R\;{p_{i,j}\left( {r,\theta} \right)}}}} \\{= {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{\alpha_{i,j}(r)}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 27} \right)\end{matrix}$wherein α_(i,j)(r) are representations of p_(i,j)(r,θ) upon averagedrotation.

The next step is to classify the p_(i,j) according to the producedα_(i,j):

For uniform α_(i,j). These include the set {p⁰}_(n=1) ^(n) ⁰ defined asall p_(i,j) corresponding to trivial (zero value) α_(i,j); and the set {p}_(n=1) ^(n) defined as all p_(i,j) corresponding to uniform andnonzero α_(i,j) (FIG. 11A). Even if trivial, these modes will beincluded as part of the uniformity producing subspace in the final stageas will be presented further in detail.

For non-uniform α_(i,j). The set {{circumflex over (p)}}_(n=1)^({circumflex over (n)}) is defined as all p_(i,j) corresponding tonon-uniform and nonzero α_(i,j) shown in FIG. 11B. This is the subspacethat contains all non-uniformity producing profiles and some uniformityproducing profiles requiring further analysis to separate them. Therotationally averaged modes corresponding to {circumflex over (p)} are{circumflex over (α)}.

Uniformity Producing Profiles in Span {{circumflex over (p)}}

The sequence of functions α^(V) is defined asα^(V)=span{{circumflex over (α)}}  (Eq. 28)

The set of α^(V) is computed using the Singular Value Decompositionprocedure. The Gram-Schmidt, or any other orthogonalization procedure isapplied to remove redundant {circumflex over (α)}. The process iswritten in matrix form[α_(n) ^(V)Σ]={circumflex over (α)}^(T)Vα^(V)=V^(t){circumflex over (α)}  (Eq. 29)where the α^(V) are the orthogonal basis functions α_(n) ^(V) scaled bytheir corresponding singular value σ_(n). The goal is to determine thesubspace of span{{circumflex over (α)}} corresponding to perfectlyuniform profiles under rotation. The search for a result may be computedusing the following sequence of operations:

1. Determine the coefficients b by projecting the α_(m) ^(V) onto aperfectly uniform rotationally averaged profile with numerical value 1:

$\begin{matrix}{ɛ = {\begin{matrix}\min \\b\end{matrix}{{{\sum\limits_{j}{b_{j}\alpha_{j}^{V}}} - 1}}}} & \left( {{Eq}.\mspace{14mu} 30} \right)\end{matrix}$

2. For ε smaller than a tolerance given by physical grounds related tothe specific uniformity control problem, using the computed b, thestalled wafer profiles that give uniform films under rotation arereconstructed usingb^(T)α^(V)=b^(T)Vα_(un)  (Eq. 31)

Replacing each α_(m) in the Eq. 31 with its corresponding {circumflexover (p)}_(m) results in the uniformity producing mode {circumflex over(β)}₀:{circumflex over (β)}₀=b^(T) V ^(T){circumflex over (p)}  (Eq. 32)

3. For the null space of V^(T), there may be a corresponding stallednon-uniform profile. Therefore:0=V_(null) ^(T){circumflex over (α)} and β_(null)=V^(T)_(null){circumflex over (p)}  (Eq. 33)

4. All uniform producing modes are further combined1 Up⁰ U p Uβ₀ Uβ_(null)  (Eq. 34)

5. When producing modes that satisfy the error tolerance set for ∈cannot be longer found, the combined space containing all β modes areorthogonalized and normalized to produce the basis{β_(n)}_(n=0) ^(N)  (Eq. 35)which defines the subspace span {{circumflex over (p)}} corresponding toall non trivial stalled wafers profiles that produce perfectly uniformfilms under wafer rotation. Representative β_(n) modes are shown in FIG.12.Defining the Nearest Uniformity Producing Profile (NUPP)

Since all linear combinations of the final set of modes generate flatprofiles, the β_(n) shown in FIG. 12, may be used to generate a usefulbasis onto which a deposition profile δ(r,θ) can be projected toimmediately determine whether the particular profile will generateuniform films under rotation and, if it does not, predict the shape ofthe “nearest” profile that does. Likewise, these modes β_(n) may be usedas a part of an efficient means of optimizing the deposition process foruniformity. The NUPP, N_(u)(r,θ) is simply computed using the projectionoperation:

$\begin{matrix}{{N_{u}\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{N}{{\beta_{n}\left( {r,\theta} \right)}{\int_{0}^{2\pi}{\int_{0}^{r_{\omega}}{{\delta\left( {r,\theta} \right)}{\beta_{n}\left( {r,\theta} \right)}r{\mathbb{d}r}{\mathbb{d}\theta}}}}}}} & \left( {{Eq}.\mspace{14mu} 36} \right)\end{matrix}$

A measure of distance between the current deposition profile δ(r,θ) andits NUPP N_(u)(r,θ) can be minimized as part of a simulation-basedprocess recipe development procedure or in a run-to-run control system.

It is important to recognize that this criterion depends only on thereactor and wafer dimensions, and thus can be universally applied to anydistributed film quality to be controlled as well as any reaction andmaterial system. In other words, the present approach to uniformitycontrol can be applied to film thickness and composition (e.g., dopantlevel) control, as well as the control of other film characteristics.Furthermore, the use of the NUPP concept is also applicable in othersystems, including ion-beam etching tools where planetary motion is usedto compensate for beam divergence, or in optical coating processes whereplanetary deposition systems are used to reduce the effect ofnon-uniformities in the coating plume.

An important feature of the subject invention is the development of anunambiguous uniformity criterion that gives an optimization objectivefunction defined in terms of the minimization of the distance betweenthe “Nearest Uniformity Producing Profile” (NUPP) and the full-wafermeasurement profile.

In the original formulation of the NUPP, the uniformity optimizationcriterion Cd was defined (Eq. 23) as the centerline distance betweenδ(r,θ) and the NUPP N_(u)(r,θ), and it is this distance that isminimized as the uniformity optimization criterion.

Considering the NUPP relative to the film thickness profile shown inFIGS. 13A-13D, given that the goal is to have the NUPP and actualdeposition profile meet at r=0, a physically reasonable first choice forprocess optimization is to increase the total gas flowrate, resulting inthe deposition peak shifting to the right as the profile is forcedfurther downstream of the gas inlet due to the increased rate ofconvection. In this example, the NUPP and Δ are, for the most part,increasing functions with R for R∈Ω, and both curves have a peak valuewithin Ω. The distance between the wafer centerline values of Δ and ƒ ispositive and has a value that is a significant fraction of the mean filmthickness itself. This difference is denoted as C_(d). What emerges fromthis computation that is of immediate value is that physical intuitionsuggests the increase of the inlet gas velocity to “push” the depositionpeak towards the reactor outlet (to the right of the diagram).

Motivated by this physical insight into a potential route to improvingthickness uniformity, the solutions are computed for a range of ν_(o)and an observation is made from this sequence of simulations: when ƒ andΔ intersect at the wafer center R=R_(s), then C_(d)=0 and uniformity isprovided in the center region of the wafer. Wafer rotation and theassumption that deposition profiles are in L²(Ω) results in d δ/dr=0 atr=0 regardless of the operating conditions. However, all derivatives ofthe rotated NUPP are zero; and the effect of the intersection of thedeposition curve and the NUPP is to drive the second derivative of δwith respect to r to zero, resulting in a flat wafer profile in theneighborhood of the wafer center.

These observations have led to an unambiguous design criterion ofC_(d)=0 for improved wafer uniformity in the central region of thewafer. In the case of simulation based process optimization, operatingconditions satisfying this condition are determined simultaneously withthe simulator solution using a Newton-Raphson technique. For thisexample, the optimal inlet velocity is found to be ν_(o)=35.04 m/s. Thestalled and rotated optimal profiles are presented in FIGS. 14A-14D andclearly illustrate the improvement in uniformity.

It is important to note that the C_(d)=0 condition is not simplyequivalent to determining the operating conditions where d² δ/dr²=0 atthe wafer center. In practice it is difficult to obtain accurate valuesof derivatives of wafer measurements because of noise and other sourcesof error. Since the wafer metrology data would be projected onto arelatively small number of the least oscillatory β_(n) modes, theprojection operation has a natural noise-filtering effect, improving theaccuracy with which the criterion can be evaluated. Furthermore, inaddition to the potential physical insight into process optimizationrevealed in this analysis approach, the relative slope of theintersection Δ and ƒ curves at r=0 when C_(d)=0 may determine howquickly the films become non-uniform in the direction away from thewafer center. Likewise, the overall slope may determine how robust thefilm uniformity is to model errors and process unknowns.

An important issue to be addressed is whether optimization criteriabetter than C_(d) can be found, both in terms of improving what istheoretically possible as well as what will work best in applications tooperating reactor systems. For example, an alternate definition of thedistance to the NUPP may be to define the residual of the projection ofa film property profile δ(r,θ) onto the β_(n) modes:S(r,θ)=δ(r,θ)−N _(u)(r,θ)  (Eq. 37)and in this manner the distance to the NUPP can be computed using theweighted inner produced=

S,S

_(ρ)  (Eq. 38)where the weight function ρ could be used to focus uniformity control onrelevant regions of the wafer.

It is important to keep in mind that the NUPP is not a stationary targetprofile and changes with the operating parameters manipulated to achieveuniformity.

Based on the analysis of the uniformity optimization and controlpresented in the previous paragraphs, a thin film processing system 20is designed, which is schematically shown in FIG. 15. The system 20includes a thin film processing chamber (or reactor) 22 which includesat least one, but usually a plurality of substrates (wafers) 24 whichcan be operated either in a stalled mode or in rotating mode as shown inFIGS. 16A-16B. The wafers 24 may be carried by a susceptor 26 which alsomay rotate about its axis as needed. Formation of a thin film 28 on thewafer 24 is performed by any type of deposition, e.g. CVD, PVD, ALD,etc. The deposition of the thin film 28 can be performed on a stalledwafer 24. As deposited, a thin film 28A (FIG. 16A) is characterized by anon-uniform profile of a film quality parameter (e.g. thickness, dopantlevel, morphology, electrical properties, etc.). In order to improve theuniformity of the thin film 28A the wafer 24 is rotated to obtain thethin film 28B having a better uniformity (FIG. 16B) of its qualityparameters.

Referring back again to FIG. 15, the processing system 20 includes acomputation system 30 operatively coupled to the reactor 22bi-directionally to control process parameters 32, and/or optimizedesign parameters 34, as well as to receive measurements necessary forthe optimization procedure. For this purpose, a measuring unit 36 isemployed in the processing system 20 for measuring a quality parameterof the thin film 28 which needs to be optimized. The measuring unit 36supplies the acquired data to the computation system 30 and particularlyto the uniformity optimization and control unit 38 which is the core ofthe processing system 20 of the present invention.

A process simulator 40 is coupled to the computation system 30 which isused in an optimization subroutine of the uniformity control process ofthe present invention as will be described further in detail. Theprocess simulator 40 is a series of complex mathematical algorithmsdesigned to simulate a reactor and process conditions and permitstracking of every adjustment that is made in the system. The processsimulator also estimates how long it takes in real time to obtain therequired change. As the process simulators are known in the thin filmfabrication industry their specifics are not discussed in detail herein.

Referring to FIG. 17, representing a flow chart diagram of the softwareunderlying the operation of the uniformity optimization and control unit38, as well as the operation of the entire processing system 20 of thepresent invention, the procedure starts in block 50 “Representing Δ inTerms of a Complete Orthogonal Sequence” in which a film property Δ(x,y)which is to be made spatially uniform is defined in at least a portionof the physical domain of the reactor Ω(x, y) in accordance with Eq. 24.

The procedure further follows to the block 60 “Projecting EachOrthogonal Function Product onto the Wafer Domain,” in which adistribution profile of the film property under optimization is found inthe physical domain of the wafers, in accordance with the Eqs. 25, 26.

From block 60, the logic proceeds to block 70 “Rotation-AveragedFunctions” in accordance with Eq. 27.

Further, the logic passes to block 80 “Classifying the p_(i,j) accordingto the α_(i,j)” to separate found p_(i,j) and α_(i,j) into the subset{ρ^(o)}_(n=1) ^(n) ^(o) corresponding to trivial (zero value) α_(i,j),subset { ρ}_(n=1) ^(n) corresponding to uniform and non-zero α_(i,j),and the set of {{circumflex over (ρ)}}_(n=1) ^({circumflex over (n)})corresponding to non-uniform and non-zero α_(i,j), e.g., the subspacethat contains all non-uniformity producing profiles and some uniformityproducing profiles requiring further analysis.

The procedure further flows to the block 90 “Defining the Subspace ofspan. ({circumflex over (p)}) Corresponding to All Non-Trivial StalledSubstrate Profiles Producing Perfectly Uniform Films Under SubstrateRotation”. In this block the subroutine is performed shown in FIG. 18 tofind all modes β_(n) of the subspace of span ({circumflex over (p)}).

As shown in FIG. 18, the subroutine of the block 90 starts with“Defining the Sequence of Function α^(v)” in block 150 where the set ofα^(v) is computed using the Singular Value Decomposition procedure.Further, in block 160 “Matrix Representation of α^(V)Orthogonalization”, the orthogonizational process is applied to the setof α^(v) to reduce redundant â and the process can be written in matrixform as presented by Eq. 29.

At this point of the subroutine, performed in block 170 “Projecting theα^(v) onto a Perfectly Uniform Rotationally Averaged Profile”, thesubspace of span ({circumflex over (p)}) corresponding to perfectlyuniform profiles under rotation is determined by determiningcoefficients b by projecting the set of α_(m) ^(V) on to a perfectlyuniform rotationally averaged profile with numeric value 1 in accordancewith Eq. 30.

The procedure further follows to logic block 180 “Is ε<Tolerance?” wherethe ε is compared with a tolerance given by physical grounds related tothe specific uniformity control problem. If ε is smaller than tolerancegiven by physical grounds, the stalled wafer profiles are reconstructedin block 190 using Eq. 31. Replacing each α_(m) in Eq. 31 above with itscorresponding {circumflex over (p)}_(m) generates the uniformityproducing mode {circumflex over (β)}₀ in block 200 “{circumflex over(β)}_(o)=b^(T)V^(T){circumflex over (ρ)}” in accordance with Eq. 32.

If however in block 180 ε is not smaller than the tolerance, the logicflows to block 250 in which three strategies can take place:

1. increase truncation numbers I and/or J in the sum of the basisfunction φ_(i) and ψ_(j) defining Δ(x,y), e.g. the deposition profileover the entire susceptor, return to logical block 60, and repeat theentire procedure in blocks 60-180 to see if ε is sufficiently reduced;

2. if the strategy 1 is impossible to find sufficiently small, check tosee if any {circumflex over (β)} modes are generated by the null spaceof V_(null) ^(T) in block 210 of FIG. 18;

3. if both of the first and second strategies fail to produce any{circumflex over (β)} modes, then compute the β modes found from trivial(zero value) sub-space p^(o) and from subspace of p corresponding touniform and non-zero profiles under rotation.

The logic further flows to block 210 processing the data related to nullspace. For the null space of V^(T), there also may be a correspondingstalled non-uniform profile which would produce a uniform profile on thesubstrate under rotation. A β_(null) is found in block 210 in accordancewith Eq. 33.

From the block 210, the logic flows to block 220 “Combining AllUniformity Producing Deposition Profiles” where the space of alluniformity producing deposition profiles is formed in accordance withEq. 34.

The logic further flows to block 230 “Orthogonalizing & Normalizing” toproduce {β_(n)}_(n=o) ^(N) where the orthogalization and normalizationof the combined space from block 220 is performed to remove redundantinformation thus producing the subspace of all uniformity producingdeposition profiles β_(n), thus defining the subspace of span({circumflex over (p)}) corresponding to all non-trivial stalled waferprofiles that produce perfectly uniform films under wafer rotation. Therepresentative β_(n) modes are shown in FIG. 12.

From block 230, the procedure returns to the routine 38 of FIG. 17,particularly to block 100 “Defining NUPP” in which the NearestUniformity Producing Profile (NUPP) is computed using the projectionoperator by Eq. 36.

Particularly, in block 100, the combinations of modes β_(n) whichgenerate flat profiles on the substrate under rotation are furtherprocessed to generate a useful basis onto which a real depositionprofile can be projected to determine whether the particular realdeposition profile will generate uniform films under rotation, and, ifit does not, predict the shape of the “nearest” profile that does.

The logic further proceeds to block 110 “Determining S(r,θ)” where ameasure of distance between the current deposition profile δ and itsNUPP N_(u) is determined in accordance with Eq. 37 and further isminimized as part of a simulation-based process recipe developmentprocedure, or in a run-to-run control system. This criterion S dependsonly on the reactor and a wafer dimensions, and can be universallyapplied to any distributed film quality to be controlled, as well as anyreaction and material system.

Using the formulated optimization criterion, it is possible not only toadjust the process parameters, but also to generate reactor designs thatminimize the sensitivity of satisfying this criterion to design modeland other errors by choosing design points where satisfaction ofNUPP-based criterion is least sensitive to the most uncertain elementsof the process model used for reactor design. Determining the NUPPfunction is shown to lead to physical insight on how reactor operatingconditions should be adjusted to improve uniformity. NUPP functions withsegments that are physically infeasible (such as negative depositionrate profiles) may indicate the reactor system is operating “far” from afeasible uniformity profile.

In order to minimize the optimization criterion S, the process parameterand/or reactor design parameters are adjusted which is accomplished inblock 120. The adjusting of process parameters and/or optimization ofreactor design in order to minimize S(r,θ) may be made in several ways.For example an approach to run-to-run control and optimization isdeveloped which is based on a value of the thin film quality parameterunder optimization:

1. measuring the desired film property of a wafer operated in stalledmode and determining the corresponding NUPP;

2. computing the sensitivity of both the most recent stalled-waferprofile and the corresponding NUPP with respect to the manipulatedparameters using a process simulator to establish a search direction;

3. performing a line search (quantitative alteration of the parameter)along the computed direction, again using the simulator to determine thegreatest reduction of the distance to the NUPP; and

4. update the process recipe accordingly.

Alternatively, the control problem can be posed as the nonlinearprogramming problem:

$\begin{matrix}{\min\limits_{q}{{{\delta\left( {r,\theta} \right)} - {N_{u}\left( {r,\theta} \right)}}}} & \left( {{Eq}.\mspace{14mu} 39} \right)\end{matrix}$subject to the constraintsq≧qminq≦qmaxRes(δ(r,θ))=0  (Eq. 40)where Res is the residual function(s) of the process simulator.

To implement the uniformity optimization approach in a run-to-runcontrol framework, planetary reactor systems featuring multiplesatellite wafers may be run with a single stalled wafer that issacrificed to obtain the measurements necessary to tune the tool to thedesired operating point and to maintain its performance against driftand other disturbances. A single-wafer reactor uniformity control isachieved by operating the reactor in stalled wafer mode—this wafer'sNUPP is computed and corrective control is applied to wafers that aresubsequently processed in a rotating wafer mode. A stalled wafer isprocessed again only when the controlled film property falls out ofspecification or a major process change is implemented (e.g., processrecipe adjustment or physical reactor modification).

Alternatively, a reactor susceptor design may have a recessed position42 for a sacrificed member 44 (made as a strip or in any otherappropriate shape) of the same material as the wafers 24 (as shown inFIG. 19). The advantage of this design is that all of the wafers 24 canbe used for production and only a relatively small area is needed forthe “monitoring strip.” In this manner, expensive wafers are notsacrificed for measurements.

As presented in the subject patent application, the new approach touniformity control in semiconductor and other thin film processes isdeveloped for systems where substrates (e.g., wafers, opticalcomponents, etc.) rotation is used to improve uniformity. Based on apurely geometrical basis, the algorithm developed identifies allnon-rotating substrate deposition profiles that result in uniformprofiles under rotation. This permits the identification of the NearestUniformity Producing Profile (NUPP) of any given substrate profile,opening the door to a new approach to uniformity control. The newtechnique makes use of all available information (thus overcoming theinherent information loss that occurs as a result of the wafer rotationon post processing measurements). Finally, the proposed optimization andcontrol approach to thin film uniformity improves the ad hoc andotherwise simplistic algorithms currently used to make use of full wafermaps. Planetary reactors, while modeled extensively and in detail, havenot made use of such sophisticated run-to-run control strategies.

The NUPP-based uniformity control approach is applicable to anyuniformity criterion in a wide range of thin film processing.Optimization, control, and design applications, including all CVD, etch,PVD, ALD, as well as other thin film processes with rotating substrate,such as, for example, film processing in semiconductor, optoelectronic,and optical coating industries benefit from this approach.

Although this invention has been described in connection with specificforms and embodiments thereof, it will be appreciated that variousmodifications other than those discussed above may be resorted towithout departing from the spirit or scope of the invention as definedin the appended claims. For example, equivalent elements may besubstituted for those specifically shown and described, certain featuresmay be used independently of other features, and in certain cases,particularly locations of elements may be reversed or all withoutdeparting from the spirit or scope of the invention as defined in theappended claims.

1. A method for improving uniformity of thin films formed in a thin filmprocessing system, comprising the steps of: providing a thin filmprocessing system including at least one substrate operated in a stalledsubstrate mode and in a rotating substrate mode, said film processingsystem having process parameters and reactor design parameters;providing a computer processing system operationally coupled to saidfilm processing system; identifying, in said computer processing system,at least one first distribution profile of at least one parameter of athin film formed on said at least one substrate in said stalledsubstrate mode, said at least one first distribution profile producing auniform second distribution profile of said thin film upon said at leastone substrate rotation in said rotating substrate mode; computing, insaid computer processing system, a subspace of basis functions β_(n)corresponding to said at least one first distribution profile in saidstalled substrate mode generating said uniform second distributionprofile in said rotating substrate mode; formulating, in said computerprocessing system, a uniformity optimization criterion defined as adeviation of a deposition profile of said at least one parameter of thethin film formed on said at least one substrate in said stalledsubstrate mode thereof from said subspace of said basis functions β_(n);and optimizing said film processing system to minimize said uniformityoptimization criterion; wherein the step of computing is followed by thestep of: forming, in said computer processing system, a NearestUniformity Producing Profile (NUPP) N_(u)(r,θ) based on said subspace ofbasis function β_(n), wherein${N_{u}\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{N}{{\beta_{n}\left( {r,\theta} \right)}{\int_{0}^{2\pi}{\int_{0}^{r_{\omega}}{{\delta\left( {r,\theta} \right)}{\beta_{n}\left( {r,\theta} \right)}r{\mathbb{d}r}{\mathbb{d}\theta}}}}}}$wherein β_(n)(r,θ) is said basis function corresponding to said at leastone first distribution profile, δ(r,θ) is said deposition profile ofsaid at least one parameter of the thin film formed on said at least onesubstrate in said stalled substrate mode, and r and θ are parameters ofsaid at least one substrate physical domain ω(r,θ); and calculating saiduniformity optimization criterion S(r,θ)=δ(r,θ)−N_(u)(r,θ) at apredetermined region of said thin film formed on said at least onesubstrate.
 2. The method of claim 1, wherein the step of identifyingincludes the steps of: projecting a sequence of orthogonal completefunctions representing Δ(x, y) onto said physical domain ω(r,θ) of saidat least one substrate, wherein Δ(x, y) represents said at least oneparameter of said thin film, thus defining the deposition profile δ(r,θ)of said at least one parameter of the thin film formed on said at leastone substrate is said stalled substrate mode, wherein${\delta\left( {r,\theta} \right)} = {\sum\limits_{i,j}^{I,J}{a_{i,j}{p_{i,j}\left( {r,\theta} \right)}}}$wherein p_(i,j)(r,θ) are representation of said deposition profileδ(r,θ) over said physical domain ω(r,θ) of said at least one substrate,and α_(i,j) are contribution coefficients; determining rotation—averagedfunctions δ(r,θ) corresponding to said δ(r,θ),${{{wherein}\mspace{14mu}{\overset{\_}{\delta}\left( {r,\theta} \right)}} = {{R\;{\delta\left( {r,\theta} \right)}} = {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{\alpha_{i,j}(r)}}}}},$wherein R is a rotationally—averaging operator, and α_(i,j)(r) arerepresentations of rotationally averaged functions corresponding to saidp_(i,j)(r,θ); and selecting a subset {{circumflex over (p)}}_(n=1)^({circumflex over (n)}) of all said p_(i,j)(r,θ) corresponding tonon-uniform and non-zero {circumflex over (α)}_(i,j)(r) from saidα_(i,j)(r) for further computation.
 3. The method of claim 2, where thestep of selecting a subset is followed by the steps of: defining asequence of functions α^(V)=span{{circumflex over (α)}}; orthogonalizingthe α^(V); determining a subspace span {{circumflex over (p)}}corresponding to perfectly uniform profiles α_(i,j)(r) under rotation;and establishing uniformity producing modes {β_(n)}_(n-0) ^(N) of saidsubspace span {{circumflex over (p)}} corresponding to all non-zeroprofiles of said at least one parameter of said thin film formed on saidat least one substrate in said stalled substrate mode thereof producinguniform thin film upon rotation of said at least one substrate.
 4. Themethod of claim 3, wherein the step of selecting a subset furtherincludes the steps of: forming a subset {p⁰}_(n=1) ^(n) ⁰ of allp_(i,j)(r,θ) corresponding to zero value α_(i,j)(r); and forming asubset { p}_(n=1) ^(n) of all p_(i,j)(r,θ) corresponding to uniform andnon-zero α_(i,j)(r).
 5. The method of claim 3, wherein said at least oneparameter of the thin film is selected from the group consisting ofthickness, composition, dopants level, microstructure, electricalproperties, and morphology.
 6. The method of claim 1, wherein the stepof optimizing includes adjusting said process parameters, wherein saidcomputer processing system generates process adjustment data.
 7. Themethod of claim 1, wherein the step of optimizing includes optimizingsaid reactor design parameters, wherein said computer processing systemgenerates reactor design optimization data.
 8. The method of claim 1,wherein said thin film processing system is selected from the groupconsisting of chemical vapor deposition (CVD) systems, physical vapordeposition (PVD) systems, atomic layer deposition (ALD) systems, andetching systems.
 9. The method of claim 1, wherein the step ofoptimizing said thin film processing system comprises the steps of:measuring said at least one parameter of said thin film subsequent todeposition of said thin film; determining said NUPP for said thin film;computing sensitivity of said at least one parameter distributionprofile and said NUPP using a process simulator, thereby establishing anoptimization direction; performing a quantative alteration of said filmprocessing system in said optimization direction to obtain a greatestreduction of said uniformity criterion; and adjusting said filmprocessing system to improve the uniformity of said at least oneparameter to produce said uniform second distribution profile of thethin film in said rotating substrate mode.
 10. The method of claim 9,wherein said at least one parameter is measured on said thin film formedon said at least one substrate in said stalled substrate mode.
 11. Themethod of claim 9, wherein said thin film processing system includes asusceptor carrying said at least one substrate, and wherein the step ofoptimizing said thin film processing system further includes the stepsof: forming a recess at said susceptor; positioning a sacrificial memberin said recess, said sacrificial member being formed of a materialidentical to the material of said at least one substrate; and obtaininga value of said at least one parameter of said thin film by measuringsaid at least one parameter of the thin film formed on said sacrificialmember.
 12. A method for improving uniformity of thin films formed in athin film processing system, comprising the steps of: providing a thinfilm processing system including at least one substrate operated in astalled substrate mode and in a rotating substrate mode; providing acomputer processing system operationally coupled to said thin filmprocessing system; identifying, in said computer processing system, atleast one first distribution profile of at least one parameter of a thinfilm formed on said at least one substrate in said stalled substratemode, said at least one first distribution profile producing a uniformsecond distribution profile of said thin film upon said at least onesubstrate rotation in said rotating substrate mode; computing, in saidcomputer processing system, a subspace of basis functions β_(n)corresponding to said at least one first distribution profile in saidstalled substrate mode generating said uniform second distributionprofile in said rotating substrate mode; forming a Nearest UniformityProducing Profile (NUPP) N_(u)(r,θ) based on said subspace of basisfunction β_(n) following the step of computing, wherein${N_{u}\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{N}{{\beta_{n}\left( {r,\theta} \right)}{\int_{0}^{2\pi}{\int_{0}^{r_{\omega}}{{\delta\left( {r,\theta} \right)}{\beta_{n}\left( {r,\theta} \right)}r{\mathbb{d}r}{\mathbb{d}\theta}}}}}}$wherein β_(n)(r,θ)is said basis function corresponding to said at leastone fist distribution profile, δ(r,θ) is a deposition profile of said atleast one parameter of the thin film formed on said at least onesubstrate in said stalled substrate mode, and r and θ are parameters ofsaid at least one substrate physical domain ω(r,θ); formulating, in saidcomputer processing system, a uniformity optimization criterionS(r,θ)=δ(r,θ)−N_(u)(r,θ) defined as a deviation of the depositionprofile δ(r,θ) of said at least one parameter of the thin film formed onsaid at least one substrate in said stalled mode thereof from said NUPPN_(u)(r,θ) at a predetermined region of said thin film; and minimizingsaid uniformity optimization criterion S(r,θ).
 13. The method of claim12, wherein said step of identifying includes the steps of: projecting asequence of orthogonal complete functions representing Δ(x, y) onto saidphysical domain ω(r,θ) of said at least one substrate, wherein Δ(x, y)represents said at least one parameter of said thin film, thus definingthe deposition profile δ(r,θ) of said at least one parameter of the thinfilm formed on said at least one substrate is said stalled substratemode:${\delta\left( {r,\theta} \right)} = {\sum\limits_{i,j}^{I,J}{a_{i,j}{p_{i,j}\left( {r,\theta} \right)}}}$wherein p_(i,j)(r,θ) are representation of said deposition profileδ(r,θ) over said physical domain ω(r,θ) of said at least one substrate,and α_(i,j) are contribution coefficients; determining rotation—averagedfunctions δ(r,θ) corresponding to said δ(r,θ),${{{wherein}\mspace{14mu}{\overset{\_}{\delta}\left( {r,\theta} \right)}} = {{R\;{\delta\left( {r,\theta} \right)}} = {\sum\limits_{i,{j = 1}}^{I,J}{a_{i,j}{\alpha_{i,j}(r)}}}}},$wherein R is a rotationally—averaging operator, and α_(i,j)(r) arerepresentations of rotationally averaged functions corresponding to saidp_(i,j)(r,θ); forming a subset {{circumflex over (p)}}_(n=1)^({circumflex over (n)}) of all said p_(i,j)(r,θ) corresponding tonon-uniform and non-zero {circumflex over (α)}_(i,j)(r) from saidα_(i,j)(r) for further computation; forming a subset {p⁰}_(n=1) ^(n) ⁰of all p_(i,j)(r,θ) corresponding to zero value α_(i,j)(r); forming asubset { p}_(n=1) ^(n) of all p_(i,j)(r,θ) corresponding to uniform andnon-zero α_(i,j)(r); defining a sequence of functions α^(V)=span{α};orthogonalizing the α^(V); determining a subspace span {{circumflex over(p)}} corresponding to perfectly uniform second profiles α_(i,j)(r)under rotation; and establishing uniformity producing modes{β_(n)}_(n-0) ^(N) of said subspace span {{circumflex over (p)}}corresponding to all non-zero profiles of said at least one parameter ofsaid thin film formed on said at least one substrate in said stalledmode thereof producing perfectly uniform thin film upon rotation of saidat least one substrate.